ginac/clifford.cpp

   1 /** @file clifford.cpp
   2  *
   3  *  Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include "clifford.h"
  24 #include "ex.h"
  25 #include "idx.h"
  26 #include "ncmul.h"
  27 #include "symbol.h"
  28 #include "numeric.h" // for I
  29 #include "print.h"
  30 #include "archive.h"
  31 #include "debugmsg.h"
  32 #include "utils.h"
  33
  34 #include <stdexcept>
  35
  36 namespace GiNaC {
  37
  38 GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed)
  39 GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
  40 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
  42
  43 //////////
  44 // default constructor, destructor, copy constructor assignment operator and helpers
  45 //////////
  46
  47 clifford::clifford() : representation_label(0)
  48 {
  49         debugmsg("clifford default constructor", LOGLEVEL_CONSTRUCT);
  50         tinfo_key = TINFO_clifford;
  51 }
  52
  53 void clifford::copy(const clifford & other)
  54 {
  55         inherited::copy(other);
  56         representation_label = other.representation_label;
  57 }
  58
  59 DEFAULT_DESTROY(clifford)
  60 DEFAULT_CTORS(diracone)
  61 DEFAULT_CTORS(diracgamma)
  62 DEFAULT_CTORS(diracgamma5)
  63
  64 //////////
  65 // other constructors
  66 //////////
  67
  68 /** Construct object without any indices. This constructor is for internal
  69  *  use only. Use the dirac_ONE() function instead.
  70  *  @see dirac_ONE */
  71 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
  72 {
  73         debugmsg("clifford constructor from ex", LOGLEVEL_CONSTRUCT);
  74         tinfo_key = TINFO_clifford;
  75 }
  76
  77 /** Construct object with one Lorentz index. This constructor is for internal
  78  *  use only. Use the dirac_gamma() function instead.
  79  *  @see dirac_gamma */
  80 clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
  81 {
  82         debugmsg("clifford constructor from ex,ex", LOGLEVEL_CONSTRUCT);
  83         GINAC_ASSERT(is_ex_of_type(mu, varidx));
  84         tinfo_key = TINFO_clifford;
  85 }
  86
  87 clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(indexed::unknown, v, discardable), representation_label(rl)
  88 {
  89         debugmsg("clifford constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
  90         tinfo_key = TINFO_clifford;
  91 }
  92
  93 clifford::clifford(unsigned char rl, exvector * vp) : inherited(indexed::unknown, vp), representation_label(rl)
  94 {
  95         debugmsg("clifford constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
  96         tinfo_key = TINFO_clifford;
  97 }
  98
  99 //////////
 100 // archiving
 101 //////////
 102
 103 clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 104 {
 105         debugmsg("clifford constructor from archive_node", LOGLEVEL_CONSTRUCT);
 106         unsigned rl;
 107         n.find_unsigned("label", rl);
 108         representation_label = rl;
 109 }
 110
 111 void clifford::archive(archive_node &n) const
 112 {
 113         inherited::archive(n);
 114         n.add_unsigned("label", representation_label);
 115 }
 116
 117 DEFAULT_UNARCHIVE(clifford)
 118 DEFAULT_ARCHIVING(diracone)
 119 DEFAULT_ARCHIVING(diracgamma)
 120 DEFAULT_ARCHIVING(diracgamma5)
 121
 122 //////////
 123 // functions overriding virtual functions from bases classes
 124 //////////
 125
 126 int clifford::compare_same_type(const basic & other) const
 127 {
 128         GINAC_ASSERT(other.tinfo() == TINFO_clifford);
 129         const clifford &o = static_cast<const clifford &>(other);
 130
 131         if (representation_label != o.representation_label) {
 132                 // different representation label
 133                 return representation_label < o.representation_label ? -1 : 1;
 134         }
 135
 136         return inherited::compare_same_type(other);
 137 }
 138
 139 DEFAULT_COMPARE(diracone)
 140 DEFAULT_COMPARE(diracgamma)
 141 DEFAULT_COMPARE(diracgamma5)
 142
 143 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
 144 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 145 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 146
 147 /** Contraction of a gamma matrix with something else. */
 148 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 149 {
 150         GINAC_ASSERT(is_ex_of_type(*self, clifford));
 151         GINAC_ASSERT(is_ex_of_type(*other, indexed));
 152         GINAC_ASSERT(is_ex_of_type(self->op(0), diracgamma));
 153         unsigned char rl = ex_to_clifford(*self).get_representation_label();
 154
 155         if (is_ex_of_type(*other, clifford)) {
 156
 157                 ex dim = ex_to_idx(self->op(1)).get_dim();
 158
 159                 // gamma~mu gamma.mu = dim ONE
 160                 if (other - self == 1) {
 161                         *self = dim;
 162                         *other = dirac_ONE(rl);
 163                         return true;
 164
 165                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 166                 } else if (other - self == 2
 167                         && is_ex_of_type(self[1], clifford)) {
 168                         *self = 2 - dim;
 169                         *other = _ex1();
 170                         return true;
 171
 172                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 173                 } else if (other - self == 3
 174                         && is_ex_of_type(self[1], clifford)
 175                         && is_ex_of_type(self[2], clifford)) {
 176                         *self = 4 * lorentz_g(self[1].op(1), self[2].op(1)) * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 177                         self[1] = _ex1();
 178                         self[2] = _ex1();
 179                         *other = _ex1();
 180                         return true;
 181
 182                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 183                 // (commutate contracted indices towards each other, simplify_indexed()
 184                 // will re-expand and re-run the simplification)
 185                 } else {
 186                         exvector::iterator it = self + 1, next_to_last = other - 1;
 187                         while (it != other) {
 188                                 if (!is_ex_of_type(*it, clifford))
 189                                         return false;
 190                                 it++;
 191                         }
 192
 193                         it = self + 1;
 194                         ex S = _ex1();
 195                         while (it != next_to_last) {
 196                                 S *= *it;
 197                                 *it++ = _ex1();
 198                         }
 199
 200                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 201                         *next_to_last = _ex1();
 202                         *other = _ex1();
 203                         return true;
 204                 }
 205         }
 206
 207         return false;
 208 }
 209
 210 /** Perform automatic simplification on noncommutative product of clifford
 211  *  objects. This removes superfluous ONEs, permutes gamma5's to the front
 212  *  and removes squares of gamma objects. */
 213 ex clifford::simplify_ncmul(const exvector & v) const
 214 {
 215         exvector s;
 216         s.reserve(v.size());
 217
 218         // Remove superfluous ONEs
 219         exvector::const_iterator cit = v.begin(), citend = v.end();
 220         while (cit != citend) {
 221                 if (!is_ex_of_type(cit->op(0), diracone))
 222                         s.push_back(*cit);
 223                 cit++;
 224         }
 225
 226         bool something_changed = false;
 227         int sign = 1;
 228
 229         // Anticommute gamma5's to the front
 230         if (s.size() >= 2) {
 231                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 232                 while (true) {
 233                         exvector::iterator it = next_to_last;
 234                         while (true) {
 235                                 exvector::iterator it2 = it + 1;
 236                                 if (!is_ex_of_type(it->op(0), diracgamma5) && is_ex_of_type(it2->op(0), diracgamma5)) {
 237                                         it->swap(*it2);
 238                                         sign = -sign;
 239                                         something_changed = true;
 240                                 }
 241                                 if (it == first)
 242                                         break;
 243                                 it--;
 244                         }
 245                         if (next_to_last == first)
 246                                 break;
 247                         next_to_last--;
 248                 }
 249         }
 250
 251         // Remove squares of gamma5
 252         while (s.size() >= 2 && is_ex_of_type(s[0].op(0), diracgamma5) && is_ex_of_type(s[1].op(0), diracgamma5)) {
 253                 s.erase(s.begin(), s.begin() + 2);
 254                 something_changed = true;
 255         }
 256
 257         // Remove equal adjacent gammas
 258         if (s.size() >= 2) {
 259                 exvector::iterator it = s.begin(), itend = s.end() - 1;
 260                 while (it != itend) {
 261                         ex & a = it[0];
 262                         ex & b = it[1];
 263                         if (is_ex_of_type(a.op(0), diracgamma) && is_ex_of_type(b.op(0), diracgamma)) {
 264                                 const ex & ia = a.op(1);
 265                                 const ex & ib = b.op(1);
 266                                 if (ia.is_equal(ib)) {
 267                                         a = lorentz_g(ia, ib);
 268                                         b = dirac_ONE(representation_label);
 269                                         something_changed = true;
 270                                 }
 271                         }
 272                         it++;
 273                 }
 274         }
 275
 276         if (s.size() == 0)
 277                 return clifford(diracone(), representation_label) * sign;
 278         if (something_changed)
 279                 return nonsimplified_ncmul(s) * sign;
 280         else
 281                 return simplified_ncmul(s) * sign;
 282 }
 283
 284 ex clifford::thisexprseq(const exvector & v) const
 285 {
 286         return clifford(representation_label, v);
 287 }
 288
 289 ex clifford::thisexprseq(exvector * vp) const
 290 {
 291         return clifford(representation_label, vp);
 292 }
 293
 294 //////////
 295 // global functions
 296 //////////
 297
 298 ex dirac_ONE(unsigned char rl)
 299 {
 300         return clifford(diracone(), rl);
 301 }
 302
 303 ex dirac_gamma(const ex & mu, unsigned char rl)
 304 {
 305         if (!is_ex_of_type(mu, varidx))
 306                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 307
 308         return clifford(diracgamma(), mu, rl);
 309 }
 310
 311 ex dirac_gamma5(unsigned char rl)
 312 {
 313         return clifford(diracgamma5(), rl);
 314 }
 315
 316 ex dirac_gamma6(unsigned char rl)
 317 {
 318         return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
 319 }
 320
 321 ex dirac_gamma7(unsigned char rl)
 322 {
 323         return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
 324 }
 325
 326 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 327 {
 328         varidx mu((new symbol)->setflag(status_flags::dynallocated), dim);
 329         return indexed(e, mu.toggle_variance()) * dirac_gamma(mu, rl);
 330 }
 331
 332 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 333  *  is that of a clifford object with the specified representation label. */
 334 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
 335 {
 336         return ti == (TINFO_clifford + rl);
 337 }
 338
 339 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
 340 {
 341         if (is_ex_of_type(e, clifford)) {
 342
 343                 if (ex_to_clifford(e).get_representation_label() == rl
 344                  && is_ex_of_type(e.op(0), diracone))
 345                         return trONE;
 346                 else
 347                         return _ex0();
 348
 349         } else if (is_ex_exactly_of_type(e, add)) {
 350
 351                 // Trace of sum = sum of traces
 352                 ex sum = _ex0();
 353                 for (unsigned i=0; i<e.nops(); i++)
 354                         sum += dirac_trace(e.op(i), rl, trONE);
 355                 return sum;
 356
 357         } else if (is_ex_exactly_of_type(e, mul)) {
 358
 359                 // Trace of product: pull out non-clifford factors
 360                 ex prod = _ex1();
 361                 for (unsigned i=0; i<e.nops(); i++) {
 362                         const ex &o = e.op(i);
 363                         unsigned ti = o.return_type_tinfo();
 364                         if (is_clifford_tinfo(o.return_type_tinfo(), rl))
 365                                 prod *= dirac_trace(o, rl, trONE);
 366                         else
 367                                 prod *= o;
 368                 }
 369                 return prod;
 370
 371         } else if (is_ex_exactly_of_type(e, ncmul)) {
 372
 373                 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
 374                         return _ex0();
 375
 376                 // Expand product, if necessary
 377                 ex e_expanded = e.expand();
 378                 if (!is_ex_of_type(e_expanded, ncmul))
 379                         return dirac_trace(e_expanded, rl, trONE);
 380
 381                 // gamma5 gets moved to the front so this check is enough
 382                 bool has_gamma5 = is_ex_of_type(e.op(0).op(0), diracgamma5);
 383                 unsigned num = e.nops();
 384
 385                 if (has_gamma5) {
 386
 387                         // Trace of gamma5 * odd number of gammas and trace of
 388                         // gamma5 * gamma.mu * gamma.nu are zero
 389                         if ((num & 1) == 0 || num == 3)
 390                                 return _ex0();
 391
 392                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 393                         if (num == 5)
 394                                 return trONE * I * eps0123(e.op(1).op(1), e.op(2).op(1), e.op(3).op(1), e.op(4).op(1));
 395
 396                         // Tr gamma5 gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 gamma.mu5 gamma.mu6 = ...
 397                         if (num == 7) {
 398                                 ex i1 = e.op(1).op(1), i2 = e.op(2).op(1),
 399                                    i3 = e.op(3).op(1), i4 = e.op(4).op(1),
 400                                    i5 = e.op(5).op(1), i6 = e.op(6).op(1);
 401                                 return trONE * I * (lorentz_g(i1, i2) * eps0123(i3, i4, i5, i6)
 402                                                   - lorentz_g(i1, i3) * eps0123(i2, i4, i5, i6)
 403                                                   + lorentz_g(i1, i4) * eps0123(i2, i3, i5, i6)
 404                                                   - lorentz_g(i1, i5) * eps0123(i2, i3, i4, i6)
 405                                                   + lorentz_g(i1, i6) * eps0123(i2, i3, i4, i5)
 406                                                   + lorentz_g(i2, i3) * eps0123(i1, i4, i5, i6)
 407                                                   - lorentz_g(i2, i4) * eps0123(i1, i3, i5, i6)
 408                                                   + lorentz_g(i2, i5) * eps0123(i1, i3, i4, i6)
 409                                                   - lorentz_g(i2, i6) * eps0123(i1, i3, i4, i5)
 410                                                   + lorentz_g(i3, i4) * eps0123(i1, i2, i5, i6)
 411                                                   - lorentz_g(i3, i5) * eps0123(i1, i2, i4, i6)
 412                                                   + lorentz_g(i3, i6) * eps0123(i1, i2, i4, i5)
 413                                                   + lorentz_g(i4, i5) * eps0123(i1, i2, i3, i6)
 414                                                   - lorentz_g(i4, i6) * eps0123(i1, i2, i3, i5)
 415                                                   + lorentz_g(i5, i6) * eps0123(i1, i2, i3, i4));
 416                         }
 417
 418                         // Tr gamma5 S_2k =
 419                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 420                         ex result;
 421                         for (int i=1; i<num-3; i++) {
 422                                 ex idx1 = e.op(i).op(1);
 423                                 for (int j=i+1; j<num-2; j++) {
 424                                         ex idx2 = e.op(j).op(1);
 425                                         for (int k=j+1; k<num-1; k++) {
 426                                                 ex idx3 = e.op(k).op(1);
 427                                                 for (int l=k+1; l<num; l++) {
 428                                                         ex idx4 = e.op(l).op(1);
 429                                                         vector<int> iv;
 430                                                         iv.reserve(num-1);
 431                                                         exvector v;
 432                                                         v.reserve(num-1);
 433                                                         iv.push_back(i); iv.push_back(j); iv.push_back(k); iv.push_back(l);
 434                                                         for (int n=1; n<num; n++) {
 435                                                                 if (n == i || n == j || n == k || n == l)
 436                                                                         continue;
 437                                                                 iv.push_back(n);
 438                                                                 v.push_back(e.op(n));
 439                                                         }
 440                                                         int sign = permutation_sign(iv);
 441                                                         result += sign * eps0123(idx1, idx2, idx3, idx4)
 442                                                                 * dirac_trace(ncmul(v), rl, trONE);
 443                                                 }
 444                                         }
 445                                 }
 446                         }
 447                         return result * I;
 448
 449                 } else { // no gamma5
 450
 451                         // Trace of odd number of gammas is zero
 452                         if ((num & 1) == 1)
 453                                 return _ex0();
 454
 455                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 456                         if (num == 2)
 457                                 return trONE * lorentz_g(e.op(0).op(1), e.op(1).op(1));
 458
 459                         // Tr gamma.mu gamma.nu gamma.rho gamma.sig = 4 (g.mu.nu g.rho.sig + g.nu.rho g.mu.sig - g.mu.rho g.nu.sig
 460                         if (num == 4)
 461                                 return trONE * (lorentz_g(e.op(0).op(1), e.op(1).op(1)) * lorentz_g(e.op(2).op(1), e.op(3).op(1))
 462                                               + lorentz_g(e.op(1).op(1), e.op(2).op(1)) * lorentz_g(e.op(0).op(1), e.op(3).op(1))
 463                                               - lorentz_g(e.op(0).op(1), e.op(2).op(1)) * lorentz_g(e.op(1).op(1), e.op(3).op(1)));
 464
 465                         // Traces of 6 or more gammas are computed recursively:
 466                         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 467                         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 468                         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 469                         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 470                         //   - ...
 471                         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 472                         exvector v(num - 2);
 473                         int sign = 1;
 474                         const ex &ix1 = e.op(0).op(1);
 475                         ex result;
 476                         for (int i=1; i<num; i++) {
 477                                 for (int n=1, j=0; n<num; n++) {
 478                                         if (n == i)
 479                                                 continue;
 480                                         v[j++] = e.op(n);
 481                                 }
 482                                 result += sign * lorentz_g(ix1, e.op(i).op(1)) * dirac_trace(ncmul(v), rl, trONE);
 483                                 sign = -sign;
 484                         }
 485                         return result;
 486                 }
 487         }
 488
 489         return _ex0();
 490 }
 491
 492 ex canonicalize_clifford(const ex & e)
 493 {
 494         if (is_ex_exactly_of_type(e, add)) {
 495
 496                 ex sum = _ex0();
 497                 for (unsigned i=0; i<e.nops(); i++)
 498                         sum += canonicalize_clifford(e.op(i));
 499                 return sum;
 500
 501         } else if (is_ex_exactly_of_type(e, mul)) {
 502
 503                 ex prod = _ex1();
 504                 for (unsigned i=0; i<e.nops(); i++)
 505                         prod *= canonicalize_clifford(e.op(i));
 506                 return prod;
 507
 508         } else if (is_ex_exactly_of_type(e, ncmul)) {
 509
 510                 // Expand product, if necessary
 511                 ex e_expanded = e.expand();
 512                 if (!is_ex_of_type(e_expanded, ncmul))
 513                         return canonicalize_clifford(e_expanded);
 514
 515                 if (!is_ex_of_type(e.op(0), clifford))
 516                         return e;
 517
 518                 exvector v;
 519                 v.reserve(e.nops());
 520                 for (int i=0; i<e.nops(); i++)
 521                         v.push_back(e.op(i));
 522
 523                 // Stupid bubble sort because we only want to swap adjacent gammas
 524                 exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
 525                 if (is_ex_of_type(it->op(0), diracgamma5))
 526                         it++;
 527                 while (it != next_to_last) {
 528                         if (it[0].op(1).compare(it[1].op(1)) > 0) {
 529                                 ex save0 = it[0], save1 = it[1];
 530                                 it[0] = lorentz_g(it[0].op(1), it[1].op(1));
 531                                 it[1] = _ex2();
 532                                 ex sum = ncmul(v);
 533                                 it[0] = save1;
 534                                 it[1] = save0;
 535                                 sum -= ncmul(v);
 536                                 return canonicalize_clifford(sum);
 537                         }
 538                         it++;
 539                 }
 540                 return ncmul(v);
 541         }
 542
 543         return e;
 544 }
 545
 546 } // namespace GiNaC